13.3 Portmanteau tests
Rather than look at each \(r_k\) separately, a portmanteau test combines the first \(K\) of them into one statistic.
Proposition 13.3.1. For white noise with finite fourth moment, the sample autocovariances \(c_j\) and \(c_k\) (\(1 \le j < k\)) are uncorrelated. Hence \(r_1, \dots , r_K\) are approximately independent \(N(0, 1/n)\) for large \(n\), and \[ Q = n\sum _{k=1}^Kr_k^2 \] is approximately \(\chi ^2_K\).
Proof. With mean \(0\), \(c_j = \frac 1n\sum _tX_tX_{t+j}\) and \(c_k = \frac 1n\sum _sX_sX_{s+k}\). A term \(E(X_tX_{t+j}X_sX_{s+k})\) is non-zero only if the four indices pair up, since a variable appearing once has mean \(0\). As \(j \ge 1\), \(t \ne t + j\), so either \(t = s\) and \(t + j = s + k\), giving \(j = k\), or \(t = s + k\) and \(t + j = s\), giving \(j = -k < 0\). Neither is possible, so \(\Cov (c_j, c_k) = 0\). By Proposition 11.2.1, each \(\sqrt nr_k\) is approximately \(N(0, 1)\); jointly they are approximately normal and uncorrelated, hence approximately independent, and the sum of their squares is approximately \(\chi ^2_K\). □
Applied to the residuals of a fitted ARMA(\(p, q\)) model, \(Q\) is the Box–Pierce statistic, and it is compared with \(\chi ^2_{K-p-q}\): the degrees of freedom are reduced by the number of estimated parameters, because fitting makes the residuals look more random than true shocks would. In small samples \(\chi ^2\) is a poor approximation to the distribution of \(Q\). For normal white noise, \(\Var (r_k)\) is closer to \((n - k)/(n(n + 2))\) than to \(1/n\), and dividing each \(r_k^2\) by this variance gives the Ljung–Box statistic, \[ Q^* = n(n + 2)\sum _{k=1}^K\frac {r_k^2}{n - k} , \] also compared with \(\chi ^2_{K-p-q}\), and now the standard choice. A large value of \(Q^*\) means the residuals are autocorrelated and the model is inadequate. \(K\) is usually taken between about \(10\) and \(20\).
Overfitting
Another check is to fit a deliberately larger model, for example AR(\(p + 1\)) when AR(\(p\)) was identified, or ARMA(\(p, q + 1\)), and see whether the extra parameter is significantly different from zero (its estimate is more than about two standard errors from zero) and whether the original estimates change much. If not, the original model is adequate.
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